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21
Scalar curvature for the noncommutative two torus
 J. Noncommut. Geom
"... Abstract. The scalar curvature for the noncommutative four torus T4Θ, where its flat geometry is conformally perturbed by a Weyl factor, is computed by making the use of a noncommutative residue that involves integration over the 3sphere. This method is more convenient since it does not require the ..."
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Abstract. The scalar curvature for the noncommutative four torus T4Θ, where its flat geometry is conformally perturbed by a Weyl factor, is computed by making the use of a noncommutative residue that involves integration over the 3sphere. This method is more convenient since it does not require the rearrangement lemma and it is advantageous as it explains the simplicity of the final functions of one and two variables, which describe the curvature with the help of a modular automorphism. In particular, it readily allows to write the function of two variables as the sum of a finite difference and a finite product of the one variable function. The curvature formula is simplified for dilatons of the form sp, where s is a real parameter and p ∈ C∞(T4Θ) is an arbitrary projection, and it is observed that, in contrast to the two dimensional case studied by A. Connes and H. Moscovici, unbounded functions of the parameter s appear in the final formula. An explicit formula for the gradient of the analog
LeviCivita’s theorem for noncommutative tori
"... Abstract. We show how to define Riemannian metrics and connections on a noncommutative torus in such a way that an analogue of LeviCivita’s theorem on the existence and uniqueness of a Riemannian connection holds. The major novelty is that we need to use two different notions of noncommutative vec ..."
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Cited by 8 (0 self)
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Abstract. We show how to define Riemannian metrics and connections on a noncommutative torus in such a way that an analogue of LeviCivita’s theorem on the existence and uniqueness of a Riemannian connection holds. The major novelty is that we need to use two different notions of noncommutative vector field. LeviCivita’s theorem makes it possible to define Riemannian curvature using the usual formulas.
A RiemannRoch theorem for the noncommutative two torus
 Journal of Geometry and Physics
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THE CURVATURE OF THE DETERMINANT LINE BUNDLE ON THE NONCOMMUTATIVE TWO TORUS
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, 2015
"... We extend the canonical trace of Kontsevich and Vishik to the algebra of noninteger order classical pseudodifferntial operators on noncommutative tori. We consider the spin spectral triple on noncommutative tori and prove the regularity of eta function at zero for the family of operators eth=2Deth= ..."
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We extend the canonical trace of Kontsevich and Vishik to the algebra of noninteger order classical pseudodifferntial operators on noncommutative tori. We consider the spin spectral triple on noncommutative tori and prove the regularity of eta function at zero for the family of operators eth=2Deth=2 and the coupled Dirac operator D+A on noncommutative 3torus. Next, we consider the conformal variations of D(0) and we show that the spectral value D(0) is a conformal invariant of noncommutative 3torus. Next, we study the conformal variation of ′jDj(0) and show that this quantity is also a conformal invariant of odd noncommutative tori. This the analogue of the vanishing of the conformal anomaly of LogDet in odd dimensions in commutative case. We also
SEE PROFILE
, 2016
"... All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately. Available from: L. Dabrowski ..."
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All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately. Available from: L. Dabrowski
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 071, 9 pages LeviCivita’s Theorem for
"... Abstract. We show how to define Riemannian metrics and connections on a noncommutative torus in such a way that an analogue of LeviCivita’s theorem on the existence and uniqueness of a Riemannian connection holds. The major novelty is that we need to use two different notions of noncommutative vect ..."
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Abstract. We show how to define Riemannian metrics and connections on a noncommutative torus in such a way that an analogue of LeviCivita’s theorem on the existence and uniqueness of a Riemannian connection holds. The major novelty is that we need to use two different notions of noncommutative vector field. LeviCivita’s theorem makes it possible to define Riemannian curvature using the usual formulas.
DELONE SETS AND MATERIAL SCIENCE: A PROGRAM
"... Abstract. These notes are proposing a program liable to provide physicists working in material science, especially metallic liquids and glasses, the mathematical tools they need to build an atomic scale theory of Continuous Mechanics including plasticity, fluidity and, hopefully, fractures. Using t ..."
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Abstract. These notes are proposing a program liable to provide physicists working in material science, especially metallic liquids and glasses, the mathematical tools they need to build an atomic scale theory of Continuous Mechanics including plasticity, fluidity and, hopefully, fractures. Using the long list of datas and numerical simulations accumulated during the last forty years, physicists have identified a new class of degrees of freedom, besides the elastic ones, which will be called anankeons here [7]. They are dominant in the liquid phase and they explain the properties related to plastic deformations of the solid phase. It is advocated that Delone sets provide a natural frame within which such a theory can be expressed. The use of Voronoi tiling and its dual construction, called the Delaunay triangulation, gives a discretization of the data. The concept of Pachner move or Delaunay flips permits to describe very precisely what the anankeons are. A partition of the configuration space into contiguity domains leads to a graph on which a Markov process can be built to describe the anakeon dynamics. At last, a speculative Section is giving an attempt to describe the Continuous Mechanics of a condensed material in terms of a Noncommutative Geometry of the configuration space. 1.